icpr2011

7/28/2019 icpr2011

1/436

JOURNAL OFINDIAN COUNCIL

OF PHILOSOPHICALRESEARCH

Issue on the Theme ofLogic and Philosophy Today

Guest Editors:

Amitabha Gupta and Johan van Benthem

Volume XXVIINumber 1

January-March 2010

Editor: Mrinal Miri

Executive Editor: Godabarisha Mishra

Indian Council of Philosophical Research

Darshan Bhawan

36, Tughlakabad Institutional Area, Mehrauli-Badarpur Road

New Delhi 110062

7/28/2019 icpr2011

2/436

Editorial Advisory Board

K. Ramakrishna Rao J. N. Mohanty

35, Dasappa Hills, Department of PhilosophyVisakhapatanam 530 003 Temple University, Philadelphia

USA

Michael McGhee Akeel Bilgrami

University of Liverpool Department of PhilosophyBrownlow Hill Colombia University, PhiladelphiaLiverpool, L69, New YorkUnited Kingdom USA

T. N. Madan Ashok VohraMD-6, Sah Vikas Department of Philosophy68, I. P. Extension University of DelhiDelhi 110092 Delhi

Vinit Haksar Srinivasa Rao

School of Philosophy, Psychology B-406, Gagan Vihar ApartmentsUniversity of Edinburgh Raja Rajeswari NagarEdinburgh EH8 9AD Bangalore 5600098

Articles published in this Journal are indexed in the

Philosophers Index, USA

ISSN - 0970-7794

INDIAN COUNCIL OF PHILOSOPHICAL, RESEARCH

Typeset & Printed in Indiaat Datagraph Creations Pvt. Ltd., Delhi 11052

(D. K. Fine Art Press)and Published by Member-Secretary

for Indian Council of Philosophical ResearchDarshan Bhawan

36, Tughlakabad Institutional Area

Mehrauli-Badarpur Road, New Delhi 110062

7/28/2019 icpr2011

3/436

i

Contents

AMITABHA GUPTA AND JOHAN VAN BENTHEM 1

Introduction

ACKNOWLEDGEMENTS 9

PART I History of Logic

WILFRID HODGES AND STEPHEN READ 13Western Logic

FABIEN SCHANG 47

Two Indian Dialectical Logics:saptabhang andcatus.kot.i

PRABAL K. SEN AND AMITA CHATTERJEE 77Navya-Nyaya Logic

FENRONG LIU AND WUJING YANG 101A Brief History of Chinese Logic

PART II Mathematical Logic and Foundations

ANAND PILLAY 127Model Theory

JOUKO VNNEN 143A Taste of Set Theory for Philosophers

JEREMY AVIGAD 165Understanding, Formal Verification, and

the Philosophy of Mathematics

S. BARRY COOPER 199Computability Theory

7/28/2019 icpr2011

4/436

ii

HIROAKIRA ONO 221Algebraic Logic

PART III Logics of Processes and ComputationFRANK WOLTER AND MICHAEL WOOLDRIDGE 249Temporal and Dynamic Logic

SAMSON ABRAMSKY 277Logic and Categories as Tools for Building Theories

R. RAMANUJAM 305Memory and Logic: a Tale from Automata Theory

PART IV Logics of Information and Agency

ERIC PACUIT 341Logics of Informational Attitudes and Informative Ac-

tions

RICHARD BOOTH AND THOMAS MEYER 379Belief Change

ROHIT PARIKH 413

Some Remarks on Knowledge, Games and Society

CONTRIBUTORS 427

7/28/2019 icpr2011

5/436

Logic and Philosophy Today:Editorial Introduction

AMITABHA GUPTA AND JOHAN VAN BENTHEM

The Initiative This special issue of the Journal of the Indian Councilof Philosophical Research (JICPR; http:www.icpr.injournal.html), is theresult of a recent initiative aimed at improving the interactions betweencontemporary logic and philosophy at universities and colleges in India.This initiative arose out of a chance meeting between Professors MrinalMiri, the Editor of the JICPR, and Amitabha Gupta. During that meet-ing, Professor Miri expressed his desire to bring out a Special Issue of theJICPR on the interface of recent developments in Logic and Philosophy.The Journal has maximum reach throughout the country. It was thoughtthat it would be the best instrument to disseminate knowledge of modernlogic and its relationship to philosophy in order to enhance the levels ofresearch and education of logic in India. There are already eminent andoutstanding Indian logicians residing outside India. What we need now isa strong group inside India involved in advanced research and in trainingbrilliant Indian minds, unleashing local energies in the field - as in ancienttimes with the Nyaya-Vaishes.hika, Jaina and Buddhist schools.

Eorts have already started in India to rejuvenate advanced researchand education in logic and its applications, with successful outreach intomathematics and computer science, by organizing Conferences and WinterSchools and forming a new Association for Logic in India (ALI; http:ali.cmi.ac.in), overseeing a wide range of initiatives, including scientific eventsand various publications. The initiative to publish a Special Issue of theJICPR is in line with this, complementing these eorts by specifically tar-geting the field of philosophy in India and its activities and programmesrelating to research, teaching and learning, by highlighting recent develop-ments in logic and their relevance to philosophy.

The urgent need to come up with a publication that would impact a broadphilosophy community in India by making modern logic accessible to itstruck a sympathetic chord with Professor Johan van Benthem, a logicianbased at Stanford University and the University of Amsterdam, who hasinitiated and supported the cause of propagating logic the world over, in-

1

7/28/2019 icpr2011

6/436

2

cluding recently in China, and who has been associated with the recentIndian eorts from their very inception. Thus, Gupta and van Benthemwere invited as Guest Editors entrusted with the ambitious task of bring-ing out an innovative and distinctive volume on "Logic and Philosophy

Today" of the JICPR, soliciting articles from among first-rate logicians inall continents. The volume that you are holding in your hand right nowis the result of this editorial collaboration between two Dutch and Indiancolleagues. But at the same time, it is much more than that, being the con-crete outcome of a truly international eort. It is a pleasure to note theoverwhelming response of top-ranking logicians to help enliven the inter-face of logic and philosophy in India by contributing a paper to this SpecialIssue of the JICPR. Likewise, the support of the Indian Council of Philo-sophical Research (ICPR) in Delhi, http:www.icpr.in, for this enterprisehas been generous and gracious all the way.

After this brief history and acknowledgment, let us now turn to mattersof content. What you see here before you is a lively panorama of logic re-search today in a broader setting, written by a large group of distinguishedauthors who each open a window to their field of expertise for a generalphilosophical audience. Our aim in all this is to give our readers an im-pression of what is going on, as well as a path into the literature. Let usfirst say a bit more about the intellectual background as we see it.

Logic and philosophy over time The juxtaposition of two fields in ourtitle needs no justification. There is a millennia-old history of fruitful inter-actions between logic and philosophy, in both Western and Eastern tradi-

tions. But paths have diverged in recent years. During the last half-century,modern logic has been undergoing a fast expansion of themes and new in-terdisciplinary alliances, a rich new reality that has hardly registered in theconsciousness of philosophers, even those well-disposed toward logic in-deed, even those who teach it. What we have tried to do with this issue isprovide the reader with a map of major thematic developments in modernlogic and its current interfaces.

Logic today Broadly speaking, modern logic was forged in the study ofthe foundations of mathematics, its rigour and consistency. In eect, thisconcern with truth and proof in mathematics was a contraction of the tradi-tional agenda of reasoning in general domains, still found with great 19thcentury logicians like Bolzano or Peirce. But it led to the Golden Age ofMathematical Logic with Frege, Russell, Hilbert and Gdel, whose results

7/28/2019 icpr2011

7/436

3

are still central to the discipline as we know it today. At the same time,these new technical insights turned out to be relevant to philosophy, illumi-nating old issues and creating new directions, witness the work of Wittgen-stein, Carnap, or Quine. What has happened after the Second World War is

both a continuation of these streams, with many new eminent names join-ing the pioneers, and also the rise of a wealth of new interfaces of logicwith other disciplines. These include linguistics, computer science, and inrecent years, also economics and psychology. Logical structures and meth-ods have turned out to be crucial in studying natural language, computa-tion, information flow, interaction, and above all, our cognitive abilities ingeneral. Thus, in a sense, logic is returning to its old broad agenda oncemore, but with new mathematical tools.

Migrations This broad contemporary role of logic also presents philos-

ophy with new interfaces. It would be hard to write the intellectual his-tory of major themes in logic and philosophy in the last century withouttracing their striking further intellectual migrations back and forth acrossacademia. Here is one such saga out of many. It was philosophers whostarted the study of counterfactual conditionals in their analysis of natu-ral laws; logicians then developed these ideas into conditional logics be-yond what mathematical logic provides, and this topic then turned out tobe crucial to understanding non-monotonic consequence relations for prac-tical default reasoning in artificial intelligence, while finally, the later logicsystems are now being applied in areas as far apart (to the superficial ob-server) as legal argumentation, the linguistic semantics of normality, brain

research with neural nets, and recently, even the study of traditional In-dian logic. Van Benthems paper Logic in Philosophy [H. B. Jacquette,ed., 2007, Handbook of the Philosophy of Logic, Elsevier, Amsterdam, pp.6599] discusses many further examples of this interplay between logic,philosophy and other disciplines, with key logical themes such as knowl-edge and information coming to reach from practical philosophy to gametheory and the social sciences, or dynamic theories of meaning that bridgephilosophy, linguistics and computer science.

Logic in India While the above trends make sense for logic and philoso-phy generally, there is a special interest in bringing these developments toattention in India. It may not be evident a priori why people in diverse cul-tures, with distinct pursuits, disparate convictions, divergent customs anda veritable feast of viewpoints would develop what Amartya Sen called ar-

7/28/2019 icpr2011

8/436

4

gumentative traditions and ingeniously nurture them. But they have. Andwhile there are scholarly debates about just what characterized the old In-dian study of logic, it is clear that inspired by a robust and vibrant traditionofnaturalism, India made its mark in the world history of logic, with fa-

mous names such as Akapda Gautama, Vasubandhu, Nagarjuna, and Sid-dhasena Divkara, representing a wealth of schools, in particular, Nyaya,Buddhist Logic, Navya Nyaya, and Jainist logic.

When modern Western logic came to India, scholars first took the Frege-Russell stance, interpreting and reformulating traditional Indian logic tofit that mould, even when the linguistic realities of Sanskrit needed to betwisted occasionally. Whether biased or not, these studies did provide thefirst significant links, and thereby started a potential conversation acrosstraditions. A later generation of distinguished scholars, influenced moreby Quine, then produced much more sensitive analyses of Indian logicalthought, widening the contacts. This volume contains a paper by PrabalSen and Amita Chatterjee, illustrating this by reviewing Navya-Nyya Logicand explaining its dicult ideas and terminology in an accessible fash-ion, using first order language in the tradition of Sibajiban Bhattacharyya,Daniel Ingalls, Bimal Krishna Matilal, Frits Staal, and in particular, Jonar-don Ganeri. In recent years, we see a third wave of studies, many of thembringing the broader logic perspectives outlined in the above to bear on un-derstanding Indian logic. This makes sense, because now that the agendaof Western Logic itself is in flux, its openness to ideas from other traditionstends to increase. These newer perspectives on interpreting Indian texts inlogic include case-based reasoning developed by Jonardon Ganeri, para-consistent logic by Graham Priest, non-monotonic logic by Claus Oetke,dialogical logic by Shahid Rahman, or modern situational logics of infor-mation flow, games, and social software by Sarah Uckelman. Our col-lection includes a paper adding yet one more perspective; Fabien Schangsurveys two Indian dialectical traditions and shows how the ancient Indianlogicians successfully buttressed the dialectic tradition.

We see in all these phases of contacts historically important stages inincreasing mutual understanding between traditions, and we hope that thisissue will encourage such studies even further.

Contents of this issue In designing this issue, we have chosen a num-ber of broad areas that allowed us to sample major developments, someextending proven classical lines, others opening new ones. Even so, thispublication is not a textbook, but an invitation. Each chapter consists of a

7/28/2019 icpr2011

9/436

5

description of an area, with some special highlights, and pointers to furtherliterature. If an author has succeeded in getting you interested, you willthen know where to look further.

In Part 1, History of Logic, Wilfrid Hodges and Stephen Read give amasterly survey of Western logic, including its subsequent ramifications inArabic logic. Fabien Schang then samples the Indian tradition through thetheme of dialectical logics, while Prabal Sen and Amita Chatterjee intro-duce its major flowering in Navya-Nyaya Logic. Fenrong Liu and WujingYang then conclude with a brief history of a perhaps less-known tradition,that of Chinese logic since Antiquity.

Part 2, Mathematical Logic and Foundations, gives some essentialtechnical pillars of the field, with chapters on model theory by Anand Pil-

lay, set theory by Jouko Vnnen, proof theory and the philosophy ofmathematics by Jeremy Avigad, computability theory by Barry Cooper,and algebraic logic by Hiroakira Ono.

Part 3, Logics of Processes and Computation, charts the thriving inter-face of logic and computer science (arguably the locus of the bulk of logicresearch today), with chapters on temporal and dynamic logic by FrankWolter and Michael Wooldridge, logic and categories by Samson Abram-sky, and logic and automata theory by Ramaswamy Ramanujam.

Part 4, Logics of Information and Agency, broadens the theme of com-

putation to communication, agency, and logical structures in social orga-nization. Eric Pacuit describes logics of informational attitudes and infor-mative actions, Richard Booth and Tommie Meyer survey modern logicsof belief change (the engine of learning and adaptation), and Rohit Parikh,the originator of the well-known program of Social Software employinglogic to understand (and improve) social procedures, ends with a key pieceon knowledge, games and society.

While many of the earlier pieces are of great relevance to philosophersinterested in logical analyis, Part 5, Logic and Its Interfaces with Philos-ophy, tells a more explicit story of contacts between logic and philosophytoday. Out of a large set of possible topics, we have selected a representa-tive sample from philosophy of language (Isidora Stojanovic), formal epis-temology (Jerey Helzner and Vincent Hendricks), logic and philosophy

7/28/2019 icpr2011

10/436

6

of science (Bas van Fraassen), logic and ethics (Sven Ove Hansson), quan-tified modal logic (Horacio Arlo Costa), logic and philosophy of mathe-matics (Hannes Leitgeb), and logic and metaphysics (Edward Zalta).

We continue this exploration, in line with what we said about migrationsearlier, with a number of congenial further interfaces in Part 6, Logic andOther Disciplines. Its chapters cover logic and quantum physics (SonjaSmets), logic and probability (Kenny Easwaran), logic and argumentationtheory (Dov Gabbay), logic and cognitive science (Alistair Isaac and JakubSzymanik), decision and game theory (Olivier Roy), and many-valued andfuzzy logics (Petr Hajek).

Taken together, the articles in our issue paint a very broad picture of ourfield. But pictures arise as much from omitting as applying brush strokes.We could have included many more topics, and we may, in later extensionsof this issues. But for now, the material presented here should be enoughto open anyones eyes to the power, sweep and beauty of logic today.

Conclusion This volume does not stand in a vacuum. Indian logicianstoday are active in university departments of mathematics, computer sci-ence, and philosophy and they have been remarkably active in recent yearsin joining the international community. Organizational eorts began with

a series of successful Conferences (2005 and 2007) and Winter Schools(2006) held at IIT Bombay on Logic and its Relationship with other Dis-ciplines that are documented in two forthcoming books: Proof, Compu-tation, and Agency: Logic at the Crossroads, Vol. 1, Amitabha Gupta,Rohit Parikh and Johan van Benthem, eds., and Games, Norms, and Rea-sons: Logic at the Crossroads Vol. 2, Johan van Benthem, Eric Pacuit andAmitabha Gupta, eds., both published by Springer Verlag.

Our present initiative hopes to strengthen this process by drawing inmore of the Indian philosophical community than was done so far, boththrough the papers in our volume and an associated meeting in a Confer-ence Week on Logic to be held at the University of Delhi from January 511, 2011. We plan to bring together our authors with teachers, researchscholars and students from Departments of Philosophy in the country aswell as participants of ALI Winter School.

7/28/2019 icpr2011

11/436

7

But let content have the final say. The various contributions in this issuepaint a rich picture of logic today, in a way that we hope will be of interestto philosophers. It has amazed us to see how easy it was to collect a distin-guished galaxy of both senior and junior logicians from all over the world,

willing to share their ideas and insights with a broader audience. The arti-cles collected here may not all be easy reads, but if you make the e ort,they will show you something that is rare: both the broader vision of to-days researchers on their broader areas, and their enthusiasm about specificthemes. Indeed, the editors themselves have learnt a lot of new things aboutlogic today, beyond what they imagined. Of course, not all our authors willagree on what modern logic is exactly, or where it is heading. We statedour own view in the above, but that was just an editorial license: takentogether, it is the papers in this volume that tell the real story of the field to-day. But no matter how one construes the march of history, we are certainthat, once these contacts have been made, Indian logicians will come to be

noticed more and more at the world-wide stage, adding original insightsin philosophy, mathematics, language, computation, and even the socialsciences. And we would not be surprised at all if some of this innovationwould come about by drawing upon Indias own rich logical tradition.

Amitabha Gupta and Johan van BenthemOctober 2, 2010

7/28/2019 icpr2011

12/436

7/28/2019 icpr2011

13/436

Acknowledgement

We would like to acknowledge the contributions of many people who helpedproduce this volume. We have already mentioned our indebtedness to theEditor of JICPR and the leaders of the ICPR who initiated and supportedthis initiative right from the start.

Our Introduction also put the spotlight on our authors, the people who

provided the real content and sparkle for these volumes. But in addition,we are grateful for various other forms of essential support.

The following colleagues reviewed papers and, through their comments,helped improve overall quality and coverage:

S.D. Agashe, Alexandru Baltag, Dietmar Berwanger, Giacomo Bonanno,Marisa dalla Chiara, Paul Dekker, Igor Douven, Jan van Eijck, Peter vanEmde Boas, Jonardon Ganeri, Valentin Goranko, Siegfried Gottwald, Nir-malya Guha, Wilfrid Hodges, Wesley Holliday, Thomas Icard, Ulrich Koh-lenbach, David Makinson, Eric Pacuit, Gordon Plotkin, Henri Prakken,K. Ramasubramanian, Manuel Rebuschi, Jan-Willem Romeijn, Hans Rott,Jeremy Seligman, Keith Stenning, Raymond Turner, Albert Visser, as well

as Jan Wolenski.We also thank the type-setting team (Hari Priyadarshan and Mr. Nirmesh

Mehta) for its herculean and prodigious help in the preparation of the finalmanuscript and producing the camera ready copy of this massive documentwhile completing the work in the shortest possible time.

Finally, we thank Sunil Simon of The Institute of Mathematical Sciences(IMSc.), Chennai and CWI, Amsterdam for his quiet and ecient logisticalassistance in coordinating this complex international process.

9

7/28/2019 icpr2011

14/436

7/28/2019 icpr2011

15/436

PART I

History of Logic

7/28/2019 icpr2011

16/436

7/28/2019 icpr2011

17/436

Western LogicWILFRID HODGES AND STEPHEN READ

The editors invited us to write a short paper that draws together the mainthemes of logic in the Western tradition from the Classical Greeks to themodern period. To make it short we had to make it personal. We set out thethemes that seemed to us either the deepest, or the most likely to be helpfulfor an Indian reader.

Western logic falls into seven periods:

(1) Classical Greece (Parmenides, Plato, Aristotle, Chrysippus; 5th to1st centuries BC)

(2) The Roman Empire (Galen, Alexander, Porphyry, John Philoponus,Boethius; 1st to 7th centuries AD)

(3) The Arabs (Al-Farab, Ibn Sna, Khunaj, Qazwn; 8th century present)

(4) The Scholastics (Peter Abelard, Peter of Spain, William of Ockham,John Buridan; 12th - 15th centuries)

(5) Renaissance to Enlightenment (Ramus, Port-Royal Logic, Leibniz;15th to 18th centuries)

(6) Transitional (Boole, Peirce, Frege, Peano, Russell, Gdel, Tarski,Gentzen; 19th century mid 20th century)

(7) The modern period (mid 20th century present)

The division is rather neat; each period built on the one before it. The chiefexception to this is Arabic logic; its high point partly overlapped the be-ginning of Scholastic logic, and after the 13th century its development wasindependent of European logic. Of course all the dates are approximate,and there were many important logicians besides those named above.

We finish this paper at the end of period (6), in the mid 20th century.That period saw some major changes of paradigm in the study of logic.By the time of the Second World War those changes had worked their way

13

7/28/2019 icpr2011

18/436

14 WILFRID HODGES AND STEPHEN READ

through the system, and post-war logicians set their minds to exploiting thenew paradigms. The rest of this volume tells you how they did it.

Our thanks to Khaled El-Rouayheb, Robert Gleave, Graham Priest, KarenThomson, Johan van Benthem and an anonymous referee for various cor-

rections and suggestions. All remaining errors are our own.

1 Classical Greece

1.1 Aristotles predecessors

Early in the 5th century BC a Greek philosopher named Parmenides, wholived in the Greek colony of Elea in South Italy, published a poem calledthe Way of Truth. In the Introduction he promised his readers that theywould learn about the well-rounded truth as well as the utterly untrust-worthy common opinions (doxai) of humans. Like the Advaita Vedanta,he believed that there is only one real entity. He claimed to prove this byassuming the opposite (the untrustworthy common opinion that there ismore than one thing) and deducing a contradiction.

His arguments were embarrassingly bad. But he established several ofthe key traditions of Greek logic. First, he showed (or claimed to show) thatwe can learn new and surprising things by using methods of pure thought.The chief method that he used is known today as Proof by Contradic-tion, or Reductio Ad Absurdum (Indian prasanga, traditionally ascribed toNagarjuna in around AD 200). But although Parmenides used this method,he didnt describe it. That was left to Aristotle around 150 years later, andis one of the reasons why Aristotle is reckoned the inventor of logic.

Second, Parmenides invented the Greek tradition of devising paradoxes;in fact paradox means contrary to common opinion, as in Parmenidesuse of the word doxai above. But again it was later Greeks who first devisedparadoxes that really challenge our thinking. The first of these later Greekswas Parmenides follower Zeno of Elea, who invented several well-knownmathematical paradoxes, including Achilles and the Tortoise. Around350 BC, the Megarian logician Eubulides discovered some of the best log-ical paradoxes, including the Liar. (Am I telling the truth or lying when Isay I am now telling a lie?)

Third, he would have been horrified to know it, but Parmenides was prob-ably one of the origins of the Greek tradition of eristic, which is the artof winning arguments regardless of whether you have a good case. (Thelawyers must have had something to do with it too.) In the early 4th cen-tury the Athenian philosopher Plato wrote a number of fictional dialogues,

7/28/2019 icpr2011

19/436

Western Logic 15

mostly involving his philosophical hero Socrates. One of them, the Eu-thydemus, is an entertaining account of a performance by two itinerantexperts in eristics. Their arguments rest mostly on obvious ambiguities inwords (just as in Parmenides poem, but their ambiguities are generally

funnier). Aristotle, who was a student of Platos, wrote a book Sophisti-cal Refutations which analysed the methods of eristics. This book had ahuge influence in late 12th century Europe after the Latin translation be-came available around 1140 (and was arguably the main stimulus to thecreation of terminism and the theory of properties of terms see 4 be-low). In later Western logic, eristics survived as a kind of undercurrent;Schopenhauer wrote a textbook of it in 1831.

There is an obvious parallel between eristic argument and the jalpa de-bates described in the Nyayasutra a few centuries later, where the aim isto win by fair means or foul. Aristotle in several places (for example So-

phistical Refutations 2) gave classifications of arguments according to theirpurpose, and several of the kinds that he mentions are really kinds of de-bate. For example he mentions didactic, dialectical, examinational,contentious and rhetorical arguments. It seems that theNyayasutra clas-sification is completely independent of Aristotles; a comparison would beinteresting.

Plato made important contributions of his own to logic. He had learnedfrom Socrates that one essential ingredient of correct reasoning is to havesound and well-defined concepts. In his dialogues he developed a tech-nique of definition which is called Division. To define a class X whichinterests us, we take a class A that includes X, and we divide it into twoclearly defined parts A1 and A2, so that one of the parts, say A1, containsall ofX. Then we split A1 into two parts A11 and A12, so that one of theparts contains all of X. We carry on subdividing until we have narroweddown to a class that contains all ofX and nothing else. Then we can defineX as the class of things that are in A and in A1 and .... The fullest accountof this method is in Platos dialogue Sophist.

1.2 Aristotle

But the main breakthrough in Classical Greek logic was certainly Aristo-tles work Prior Analytics. Its contents were probably written in the thirdquarter of the 4th century BC. Aristotles works are a strange mixture ofbooks, lectures and notes, and we are often unsure that he intended to writetreatises in their present form. Nevertheless the Prior Analytics containsone of the worlds first tightly integrated formal systems, comparable in a

7/28/2019 icpr2011

20/436


way with Pad. inis description of Sanskrit. In this work Aristotle describedrules of argument, and showed how all his rules could be derived from asmall starting set. The rules were called syllogisms.

Ignoring the modal syllogisms (which are still controversial see 4.1

below), Aristotle described what were later enumerated as nineteen syllo-gisms. In the Middle Ages they were given mock-Latin names for easymemorising. (See 4.1 below.) The first and most famous syllogism wasthe one that the medievals called Barbara. As Aristotle himself presents it,it takes the form

IfC belongs to all B, and B belongs to all A, then C belongsto all A.

The letters mark places where one can put terms, i.e. (in general) nouns ornoun phrases; the same noun should be put for A at both occurrences, andlikewise with B and C. Probably he intended that dierent terms should

be put for dierent letters too. Its virtually certain that Aristotle took theidea of using letters from the Greek geometers.

For example Aristotle might write

(1) If every fisher is a hunter, and every angler is a fisher, then every angleris a hunter.

This is our example and not his; the few explicit examples that he didgive are mostly tricky cases that needed special analysis. We took the ideaof this example from Platos definition of angler in Sophist; many peoplebelieve that Aristotle first devised his argument rules through developingPlatos definitions in this kind of way. However that may be, Aristotlesnext move was to see that the validity of the argument in (1) doesnt dependon the terms that are put for the letters. We could use any terms, providedthat the resulting sentences make sense and we always use the same termfor the same letter. So he had discovered not just valid arguments but validargument forms; every argument of that form is guaranteed to be valid. Hecould have written this form as

(2) Every B is a C. Every A is a B. So every A is a C.

just as most later logicians did. Perhaps he used the roundabout phrasingC belongs to all B because he realised that he had invented a completelynew discipline, and he wanted to mark this with some new technical termi-nology.

What was most distinctive of Aristotles contribution to logic, however,was that he gave general form to two methods: the method of showing

7/28/2019 icpr2011

21/436

Western Logic 17

syllogisms to be valid, and the method of showing invalid argument formsto be invalid. The former method was to reduce all valid syllogisms to whathe called the perfect or first figure syllogisms, and ultimately to two ofthese, Barbara (as above), and Celarent:

No B is a C. Every A is a B. So no A is a C.

The other method, of showing arguments invalid, was to find replacementsfor the constituent descriptive terms, or the symbolic letters, such that thepremises are true and the conclusion false. E.g., take the argument form:

Every A is a B. Some B is a C. So some A is a C.

If we replace A by horse, B by animal and C by donkey, we cansee that the conclusion cannot follow from the premises, since it is falseand they are true.

Although Aristotle began his career as a follower of Plato, he later as-serted his independence, and for some centuries his followers (the Peri-patetics) and the Platonists formed competing schools. This rivalry gen-erated a number of myths that still survive today; you can find some ofthem on the internet. For example it was claimed that Pythagoras and Par-menides both had systems of logic, and that Plato had inherited them. Butin fact there is not the slightest evidence that Pythagoras ever had anythingto do with logic, and certainly Parmenides had nothing like a system.

1.3 Stoic Logic

Attempts by Platonists to establish a platonist logic to rival Aristotles logicnever succeeded: Aristotle had cornered all the logically worthwhile ideasin Platos work. But the later classical Greeks were fortunate in having asecond substantial theory of logic besides Aristotles, namely that of theStoics (who inherited logical insights from some earlier logicians, notablythe Megarians). The leading figure of the Stoic school was Chrysippus,who lived in the second half of the 3rd century BC. Unfortunately no com-plete logical works from this school survive though we are told thatChrysippus himself wrote over a hundred logical treatises, including sevenon the Liar Paradox. But we know enough to point to some importantinnovations by this school.

First, they invented propositional logic. Second, their notion of modal-ity was formally dierent from Aristotles. For Aristotle (at least on onereading of his rather obscure explanations), humans are necessarily ani-mals but possibly writers; the modality goes with the description. For

7/28/2019 icpr2011

22/436


the Stoics, necessity and possibility are properties of whole assertions: Itis night is possibly the case but not necessarily the case. In the later ter-minology, Stoic modalities were de dicto, about something said. (And asin some Indian traditions, Stoic logicians used modal notions as properties

of propositions rather than as parts of propositions.) Third, Stoics had thenotion of incomplete meanings, which (to use modern terminology) havean argument place that needs to be filled. For example writes is incom-plete because it needs a subject argument, as in The moving finger writes.Fourth, they had at least the beginnings of a sophisticated philosophicaltheory of meanings, intended to answer questions like What entities aremost properly described as having a truth value? The Stoics also had areputation for being formalistic, but at this distance in time and with thescanty records that we have, it would be unsafe for us to ascribe to themany particular formalistic doctrine.

The first three of these Stoic contributions eventually passed into thegeneral practice of logic. But by the time of Arabic logic the Stoics as adistinct school of logic had faded from the record.

1.4 Acquisition of knowledge

Writers on Indian logic have often remarked that Indian logic, unlike mostmodern Western logic, is about how an individual comes to know some-thing that he or she didnt know before. Inference is a process that hap-pens in the mind of the reasoner. It is not always realised that, with onlymarginal exceptions, exactly the same was true for all proofs in Westernlogic before the beginning of the twentieth century. For example the syllo-gisms that Aristotle counted as not perfect were those where the conclu-sion doesnt obviously follow from the premises. His reductions of thesesyllogisms to perfect syllogisms were not just abstract validity proofs; theywere chains of reasoning that a reasoner could use in order to be convincedof the truth of the conclusion of a non-perfect syllogism.

One of the main purposes of logic in the West has been to validate ar-guments by bringing them to some appropriate kind of logical form. Butthis meant something dierent in traditional Western logic from what itcame to mean in the twentieth century. The traditional logicians reckonedthat a piece of informal reasoning could be reduced to steps, and each stepintroduces its own piece of knowledge. The steps could be formalised sep-arately; for example there was no requirement even to use the same termsin one step as in the next. So a complicated argument would be reduced to amixture of logical steps each simple in itself and linguistic rearrange-

7/28/2019 icpr2011

23/436

Western Logic 19

ments or paraphrases. But in the late nineteenth century logicians startedto take a very dierent approach: the terms in an argument are symbolised,and the same assignment of symbols applies to the whole argument fromstart to finish, even if the argument consists of many pages of mathemat-

ics. As a result, a modern Western student of logic learns how to operateformal proofs of much greater complexity than in the traditional format. Inthis modern style the separate steps of a formalised argument are not eachintended to convey a separate piece of knowledge at least, not in anystraightforward way. One reason for the current interest in the Scholasticobligational disputations (see 4.3 below) is that unlike syllogisms, theydo generate arguments with some significant complexity, though these ar-guments are not really proofs that give us new knowledge.

2 The Roman Empire

During the first century BC, Aristotles logical writings which had pre-viously been kept in the private hands of Peripatetics were edited andpublished as a group of books called the Organon. The editor (said to beAndronicus of Rhodes) put first the book Categories, which is about themeanings of single words. Book 2 was On Interpretation, which discussedthe ways in which words are arranged in sentences. Then he put book 3,the Prior Analytics, which explained how to arrange sentences into validarguments. Book 4, the Posterior Analytics, was about how to use syllo-gisms in order to increase our knowledge. Book 5, the Topics, was aboutdebate. Book 6 was the Sophistical Refutations; we mentioned it in 1.1

above. In one tradition, two more of Aristotles books were included in theOrganon, namely the Rhetoric and the Poetics; the first of these was aboutpersuasive public argument and the second was about the expressive forceof poetry and drama.

The Organon and other works of Aristotle contained an immense amountof learning, but they were hard to read. Around AD 200 the Peripateticphilosopher Alexander of Aphrodisias wrote commentaries on the mainworks, including the Prior Analytics. His is the first commentary to sur-vive of a tradition which lasted for a thousand years. The commentaryformat was so successful that throughout the first millennium AD and forsome while after, the main research in logic appeared in the form of com-mentaries on books of the Organon.

Students working in this tradition were shown how to break a text downinto separate inferences, and to check each inference by logic. Each infer-

7/28/2019 icpr2011

24/436


ence needed to be tickled into the shape of a syllogism by suitable para-phrasing. The result was that any substantial piece of logical analysis usedpartly logic and partly paraphrase. The paraphrase was done by intuitionbased on studying many cases, and not by rule. Leibniz later described

such steps of paraphrase as valid non-syllogistic inferences.

We can illustrate this with an important example from the Roman period.Aristotle had been interested in the nature of mathematical knowledge, andhis views about this may well have influenced later Greek mathematicalwriting, for example Euclids Elements. But it seems unlikely that the rea-soning procedures of Greek mathematics had any influence on Aristotlessyllogisms the mismatch is too great. For example most statements ingeometry use relations: lines L and Mare parallel, point p lies on line Land so on. Syllogisms had no machinery that handles relations naturally.

Nor had the propositional logic of the Stoics. The logicians of the 2ndcentury AD made the first attempts to reconcile logical methods and math-ematical ones. It was apparently Alexander of Aphrodisias who took thecrucial step of representing relations by allowing the Every and Somein syllogisms to range over pairs or triples as well as individuals.

In fact, in the 1880s C. S. Peirce took up this idea of using pairs, triplesetc. (which he credited to his own student Oscar Mitchell who had intro-duced propositions of two dimensions). On the basis of it Peirce inventedwhat we now recognise as the earliest form of first-order predicate logic.But there is an important dierence between Alexanders idea and Peirces.

Alexander never introduced any method for passing from statements aboutindividuals to statements about pairs, or from statements about pairs tostatements about triples, etc. For him and the traditional logicians who fol-lowed his lead, no such method was needed, because one could take care ofthe switch by using paraphrase between the logical steps of an argument.But Peirces predicate logic allows us to use facts about pairs to deducefacts about individuals, and so on, all within the same formalism. Todayno logician would dream of stepping outside a formal proof in mid streamin order to cover a step by paraphrasing.

The Roman Empire commentators tidied up several other aspects ofAristotles logic. One important contribution from this period was theSquare of Opposition, a diagram which records the logical relations be-tween the four propositions in the corners of the square:

7/28/2019 icpr2011

25/436

Western Logic 21

Every S is P

Some S is P Not every S is P

No S is P

contraries

subcontraries

subaltern

ate

subaltern

ate

contradictories

A proposition entails its subalternates; contraries cannot both be true, butcan be false together; subcontraries cannot be false together, but could betrue together; contradictories cannot both be true and cannot both be false.

Most of the Roman Empire commentators on Aristotle after Alexanderof Aphrodisias were in fact Platonists or Christians, not Peripatetics. Howcould they justify teaching the views of the founder of a rival philosophy?They found a tactful solution to this problem. Logic was so obviouslyvaluable that all students should learn it. But the commentators found thatthey could detach the logic from Aristotles philosophy and metaphysics.A philosophy-free logic was taught as a first step, and when the studentshad it under their belt, they would move onto the higher truths of Platonism(or later, Christianity or Islam). But the commentators didnt want to teachlogic by pure rote, so they found a kind of justification for it in semantics the study of the meanings of words and sentences. Thus the students

would learn semantics from the first two books of the Organon and thenmove on to syllogisms in the third book.

An example may help comparison with Indian traditions. The pointcomes up in various Indian treatises that when we make a deduction from ageneral rule, e.g. Whenever there is smoke there is fire, we need to pointto an instance that confirms the rule (a sadharmya-dr. s. t. anta). The RomanEmpire commentator tradition wouldnt have put it like that. If the reasonfor giving the instance is that a general rule doesnt count as true unless ithas an instance, then that should have been said in the explanation of themeaning of general rules. It should be made a point of semantics, not astep in arguments. And in fact some of the commentators of this perioddid count an armative universal statement Every A is a B as false unlessthere is at least one A. (But they allowed the negative statement No A is aB to be true when there are no As.)

7/28/2019 icpr2011

26/436


This meaning-based logic must have been the brainchild of many di er-ent scholars, but the Palestinian Platonist philosopher Porphyry of Tyre inthe late 3rd century is believed to have played a key role. Porphyry alsowrote an elementary introduction to logic; he called it the Introduction

(Eisagoge in Greek), and for several hundred years it was read by everystudent of logic. In it he mentions some philosophical problems about gen-era and species (like animal and human); these problems later becameknown as parts of the problem of universals. For example do genera andspecies really exist as entities in the world? Porphyry adds that he is delib-erately not discussing these problems. The Scholastics couldnt resist thechallenge of tackling the problem of universals, and the result was that inthe West the ideal of a philosophy-free logic went down the drain. It wasrecovered in a more scientific form through the work of Carnap, Tarski andother logicians in the period between the two world wars of the twentiethcentury. (See 6 on Tarski.)

3 The Arabs

Logic has had a good reputation through most of Islamic history. Thereare many statements in the Quran along the lines Thus do We explainthe signs in detail for those who reflect (10.24), and these are commonlyunderstood as calls to Muslims to develop their rational thinking. In theearly days of the Islamic empire there were a number of well-to-do Arabicspeakers, spread across the world from Spain to Afghanistan, who regardedskills of debate as a mark of culture. So they bought logic texts and took

lessons in logic. Tamerlane had two distinguished Arabic logicians at hiscourt in Samarqand. Ibn Sna (known in Europe as Avicenna) reportedthat in the late 990s the library of the Sultan of Bukhara (in present-dayUzbekhistan) had a room full of logical texts. Probably it contained trans-lations of most of the Roman Empire commentaries. Most of this materialis lost today, or at least uncatalogued. We know there are important Arabiclogical manuscripts that have never been edited; for example some are inTurkey and some are in the Indian National Library.

Logic did sometimes have to fight its corner. There were demarcationdisputes between logicians and linguists about which aspects of languageshould be studied in which discipline. A more serious problem developedlater: some of the main experts in logic had unorthodox religious views. Inaround 1300 Ibn Taymiyya whose religious and political writings haveinspired Osama bin Laden argued strongly against Aristotles logic. But

7/28/2019 icpr2011

27/436

Western Logic 23

about 200 years earlier Al-Ghazal had mounted a largely successful cam-paign to convince Muslims that Aristotles logic was theologically innocentand a great help for reaching the truth. Its largely thanks to Al-Ghazal thatlogic has been a major part of the madrasa syllabus ever since.

The first Arabic logician of distinction was Al-Farab in the early 10thcentury. Today philosophers cite him for his views on the relation of logicto determinacy, among other things. A century later came Ibn Sna, a log-ical giant comparable in various ways to Leibniz. It almost passes beliefthat the medieval scholars who translated classical Arabic philosophy intoLatin thought his logic was not worth translating, so that it was unknownin Europe. (But they did translate the more conservative modal logic of IbnRushd Averroes which influenced the English logicians Kilwardbyand Ockham.) One of Ibn Snas books (Easterners, which unfortunatelyis available only in an unreliable Arabic version) has a long section on howhe thought logic should be done; in comparison with Aristotles logic, this

section had much less about rules of proof, much more about how to in-terpret statements and texts, and a long section on definition. Here andelsewhere Ibn Sna emphasised that what we mean is nearly always a gooddeal more complex than what we say we mentally add conditions tothe public meanings of our public words.

Here is a typical example of Ibn Snas semantic analysis. What doesa statement Every A is (or does) B mean? If we look at examples wecan see that there are various patterns. When we say that every horse isa non-sedentary animal, we dont mean just now or sometimes, we meanalways. But again we dont imply that every horse is eternal; every horse isa non-sedentary animal for as long as it lives. But when we say God is mer-ciful, we mean it for all time. Next take the statement that everyone whotravels from Ray in Iran to Baghdad in Iraq passes through Kermanshahnear the border. A person who says this certainly doesnt mean that everysuch person passes through Kermanshah for as long as he lives; she meanssometime during the journey. On the other hand a biologist who says Ev-erything that breathes in breathes out doesnt mean that it breathes out atsome time while it was breathing in! And so on. Ibn Sna did some pre-liminary cataloguing of these and other cases. But his general position onthese examples seems to have been that we should be alert to the possibili-ties, and we should aim to reason with them in ways that we find intuitivelynatural. He believed that a training in Aristotles syllogisms would help usto do that.

Note the form of the statement about the traveller from Ray to Baghdad.It can be written as follows:

7/28/2019 icpr2011

28/436


Every traveller who makes a journey from Ray to Baghdadreaches Kermanshah during the journey.

Sentences with a similar form appeared in Scholastic logic in the early 14th

century. One due to Walter Burley reads:

Every person who owns a donkey looks at it.

Thanks to Burleys example, sentences of this kind have become knownas donkey sentences. The mark of a donkey sentence is that it containstwo parts, and the second part refers back to something introduced by animplied existential quantifier inside the first part. For first-order logic thesesentences are nonsensical: the reference in the second part is outside thescope of the quantifier. In the English-speaking world the question whatwe can infer from a donkey sentence has been seminal for research intonatural language semantics. About the same time as this research began,

Islamic jurists independently realised that they had a donkey sentence in averse of the Quran (49.6):

If a person of bad character brings you a report, you shouldscrutinize it carefully.

(Note the quantifier a report in the first part, and the back reference itin the second strictly the it is missing in the Arabic, but it is clearlyunderstood.) A number of jurists have published analyses of this verse andits implications. They make no direct reference to logic, but its plausibleto see in their analyses an indirect influence of Ibn Sna, through the logicof the madrasas. The most famous of these jurists is well known for otherreasons: Ayatollah Khomeini.

A second feature of the traveller example is that there are quantifiers bothover the traveller and over time. This makes it a proposition of two di-mensions in Oscar Mitchells sense (see 2 above). Sadly Ibn Sna had noPeirce to transmute his ideas into a radically new logic. In fact later Arabiclogicians recognised the originality of Ibn Snas examples, but often theirtendency was to introduce new moods of syllogism for each new kind ofexample. Later Arabic logicians studied further examples and duly addedfurther syllogisms. This style of logical research made strides in the Ot-toman empire during the 18th century and led to a relatively sophisticatedlogic of relations, at a time when European logic was largely moribund.

It seems to have been the Arabic logicians who began the study of rea-soning in conditions of uncertainty. Both Al-Farab and Ibn Sna repri-manded the doctor and logician Galen (2nd century AD) for missing the

7/28/2019 icpr2011

29/436

Western Logic 25

fact that most medical statements are probabilistic. Much later, in Parisin 1660, the Port-Royal Logic of Arnauld and Nicole discussed how tothink rationally about the danger of being struck by lightning. In the 19thcentury Augustus De Morgan and George Boole tried to incorporate quan-

titative probability reasoning into logic. But the trend was against them,and in the 20th century probability theory came to be recognised as a sep-arate discipline from logic. (See also the paper Logic and probability byK. Easwaran in this collection.)

One would expect some mutual influence between Arabic and Indianlogic because of the geographical closeness. But no direct influences havebeen discovered. For example one of the leading Arabic scientists, Al-Brun, being compelled to visit North India in the early 11th century aspart of the entourage of a warlord, used the opportunity to collect informa-tion on Indian science and culture. He wrote a long report with a mass of

information about Indian achievements, including philosophy and astron-omy. But his book makes no mention of Indian logic. He does refer to onelogical text, the Nyayabh asa, but he describes it as a book on Vedic inter-pretation. If he came across Indian logic at all, he simply didnt recogniseit as logic.

4 The Scholastics

Although there are important discussions of logical issues in such eleventhand early twelfth century thinkers as Anselm of Canterbury, Peter Abelardand Adam Balsham, the distinctive contribution of medieval logic as a bodyof doctrine began in the late twelfth century in the study of consequenceand fallacies. This began with the rediscovery in the Latin West of Aristo-tles doctrine of fallacy in his Sophistical Refutations (known in Latin asDeSophisticis Elenchis) and of the syllogism in his Prior Analytics. However,although Boethius (480-525) had translated all of Aristotles Organon ex-cept the Posterior Analytics (as part of a grand project, never completed, oftranslating all of Aristotles works into Latin with commentaries on them),only Boethius translations of the Categories and On Interpretation (De

Interpretatione in Latin, Peri Hermeneias in Greek) were known and incirculation at the start of the twelfth century these two were termed,along with Porphyrys Introduction, the logica vetus. During the rest of thecentury, Boethius translations of the other works emerged (from where isunknown) and in addition translations of both Analytics, the Topics and the

7/28/2019 icpr2011

30/436


Sophistical Refutations were made by James of Venice (who had studied inConstantinople) around mid-century. These became known as the logicanova. The theory of the syllogism became the basis of the medieval theoryof consequence. What is important to realise is that the assertoric syllogism

only takes up a relatively small part of the Prior Analytics. Aristotle therealso developed a theory of the modal syllogism. But whereas his theory ofthe assertoric syllogism was clear and convincing, his theory of the modalsyllogism was highly problematic.

4.1 Consequence

In fact, the syllogism is not the whole of Aristotles logic. For as we noted,Aristotles method of validating syllogisms was to reduce all syllogismsto the perfect syllogisms of the first figure and ultimately to Barbaraand Celarent. The method of reduction depended on a number of one-

premise inferences elaborated in On Interpretation, in particular, simpleconversion, conversion per accidens, subalternation, and reductio per im-

possibile. The assertoric syllogism is concerned with so-called categoricalpropositions (a better translation is predicative proposition, or subject-predicate proposition, since the Latin categorica is simply a transliterationof the Greek kategorike, predicative), in particular, the four forms EveryS is P (so-called A-propositions), No S is P (E-propositions), SomeS is P (I-propositions) and Some S is P (or better, Not every S is P,O-propositions).

One of the main sources of our knowledge of late twelfth and early thir-teenth century logic is Peter of Spain. For many centuries, he was thought

to be the same Peter of Spain as Pope John XXI, who was killed in 1276when the roof of his new library fell on him. Recently, however, it has beenestablished that this was a misidentification, and that the logician Peter wasa Dominican from Estella (Lizarra) in the Basque country. His Tractatus(Treatises) record the state of the art, and contain the famous mnemonicby which students learned the theory of the assertoric syllogism:

Barbara Celarent Darii Ferio BaraliptonCelantes Dabitis Fapesmo Frisesomorum;Cesare Camestres Festino Baroco; DaraptiFelapton Disamis Datisi Bocardo Ferison.

There are here three figures. Aristotle conceived of syllogisms as pairsof premises, asking from which such pairs a conclusion could be drawn.Those pairs of categorical propositions containing between them three terms

7/28/2019 icpr2011

31/436

Western Logic 27

could share their middle terms as subject of one and predicate of the other(figure I), as predicate of both (figure II Cesare - Baroco) and as subjectof both (figure III Darapti - Ferison). In the first figure, the predicate ofthe conclusion could be the predicate in its premise (concluding directly

Barbara - Ferio) or the subject (concluding indirectly Baralipton - Fris-esomorum). Only when the syllogism was thought of as an arrangementof three propositions, two premises and a conclusion, did it seem better tocall the indirect first figure a fourth figure, as some Stoics (e.g., Galen) andsome medievals (e.g., Buridan) did.

The mnemonic lists 19 valid syllogisms. Five more result from weaken-ing a universal conclusion by subalternation. The first three vowels givethe type of the constituent propositions; certain consonants record the re-duction steps needed to reduce the mood to a perfect syllogism, that is, onein the direct first figure. E.g., Baralipton (aai in the indirect first figure) isreduced to Barbara by converting the conclusion of Barbara per accidens

(from Every S is P to Some P is S), as indicated by the p followingthe i. The initial consonant indicates the perfect syllogism to which itreduces.

The modal syllogism results from adding one of three modalities to oneor more of the premises and the conclusion. The modalities Aristotle con-siders are necessary, possible and contingent (or two-way possible).In its full articulation, the theory was very complex. But there was some-thing puzzling right at its heart, sometimes known as the problem of thetwo Barbaras. In ch. 3 of the Prior Analytics, Aristotle says that E- andI-propositions of necessity convert simply, that is, No A is necessarily Bconverts to No B is necessarily A and Some A is necessarily B convertsto Some B is necessarily A, and necessary A-propositioins convert peraccidens, that is, Every A is necessarily B converts to Some B is neces-sarily A. But in ch. 9 of that work, he says that adding necessarily onlyto the premise of Barbara containing the predicate of the conclusion validlyyields a necessary conclusion (i.e., Every B is necessarily C, every A is B,so every A is necessarily C is valid), but not if necessarily is only addedto the other premise (i.e., Every B is C, every A is necessarily B, so every Ais necessarily C is invalid). The challenge is to find a common interpreta-tion of Every S is necessarily P which verifies these two claims. A verynatural interpretation of the remark in ch. 3 is that he takes necessity dedicto, or as the medievals would say, in the composite (or compounded)sense, or as modern logicians would say, with wide scope, so that it predi-cates necessity of the dictum, the contained assertoric proposition. But onthis reading, the modal Barbara of ch. 9 would not be valid. For necessar-

7/28/2019 icpr2011

32/436


ily every bachelor is unmarried (taken de dicto), but supposing everyonein the room is a bachelor, it does not follow that necessarily everyone inthe room is a bachelor. One way to make the syllogism valid is to take thenecessity de re, or as the medievals would say, in the divided sense, or in

modern terms, with narrow scope: every B is of necessity C. For then thesyllogism reduces to a non-modal case of Barbara with a modal predicate,of necessity C, and so is valid. But the conversions of ch. 3 fail whentaken de re.

Forcing a choice between de re and de dicto interpretations of the modalpremise may be anachronistic and out of sympathy with Aristotles meta-physical projects. Nonetheless, this and other problems with the modalsyllogism led to much discussion of modal propositions and a variety oflogics of modality in the scholastic period and among the Arabs. We willreturn to a further problem of interpretation of modal propositions shortly.

The main development of medieval logic (the logica modernorum, thelogic of the moderns, as it came to be known), however, was to developa general theory of consequence. In the twelfth century, one focus of con-cern was a claim of Aristotles, endorsed by Boethius, that no propositionentailed its contradictory, since they could not both be true, nor did anysingle proposition entail contradictories, so if it entailed one of a contradic-tory pair, it couldnt entail the other. But there is, at least with hindsight, anobvious counterexample, namely, an explicitly contradictory proposition,which entails (by the rule known as Simplification, from a conjunction toeach of its conjuncts) each of its contradictory conjuncts. Moreover, a con-tradiction entails not just its conjuncts, but any proposition whatever. Forwe can disjoin one of the contradictory conjuncts with any other proposi-tion, and since the other conjunct contradicts the first disjunct, that otherarbitrary proposition immediately follows. Such surprising results showedthat what was needed was a general theory, and it developed along twofronts. The primary line of development was a theory of inference, fram-ing inference rules in terms of the structure of the propositions in question.At the same time, the theory of fallacies developed, building on Aristo-tles theory of fallacy in his Sophistical Refutations and on his method ofcounterexamples from the Prior Analytics. In time this led to a second andsupplementary account of consequence in terms of truth-preservation.

4.2 Properties of TermsAristotle had had relatively little to say about propositional consequence inOn Interpretation apart from the rules that the later commentators incor-

7/28/2019 icpr2011

33/436

Western Logic 29

porated in the Square of Opposition (2 above). But what, for example,explains subalternation, from Every S is P to Some S is P? To explainsuch inferences, the medievals developed their distinctive theory of prop-erties of terms. As the twelfth century proceeded, many properties were

mooted: signification, supposition, appellation, copulation, ampliation, re-striction and relation were some of them. Part of the spur to this was meta-physical: if, as Aristotle had said, everything is individual, and the onlyuniversals were names, one needed a theory of signification, or meaning,to explain the functioning of names. Supposition then explained how termsfunctioned in propositions, and in particular picked out that class of thingsthe term stood for, and how it did so. Thus the theory of supposition hastwo aspects, the first concerning what the term stands for, the other themode of supposition. Sometimes, for example, a term supposits for itself,or some other term (one which it doesnt signify), as in Man is a noun,or The spoken sounds pair and pear sound the same we nowadays

mark such uses with inverted commas. The medievals said the term hadmaterial supposition. Other cases where a term does not supposit for thethings it signifies are when it stands for the universal (if there is one) or theconcept, e.g., in Man is a species. This was said to be a case of simplesupposition. Some authors, especially realists, thought supposition shouldbe restricted to subjects, and predicates had copulation (i.e., coupled to thesubject). Others thought predicates had simple supposition, for the uni-versal. The hard-line nominalists, however, like William of Ockham andJohn Buridan, in the fourteenth century, thought both subject and predicatestood for individuals. For example, in A man is running, man and run-ning stand for men and runners, respectively, and have so-called personalsupposition. The proposition is true if subject and predicate supposit forsomething in common if the class of men overlaps the class of runners.Thus subject and predicate in personal supposition stand for everything ofwhich the term is presently true. Man supposits for all (present) men andrunning for all those presently running.

What, however, of, e.g., Some young man was running? SupposeSocrates is now old, but in his youth he ran from time to time. Youngrestricts man to supposit only for young men; but was running ampli-ates the subject young man to supposit for what is now, or was at sometime, a young man. So the proposition is true of Socrates, since he was atsome time a young man and ran. Indeed, it is true if he never ran in hisyouth but ran yesterday, say. So ampliation and restriction analyse Someyoung man was running to say Something which is or was at some time ayoung man was at some time (not necessarily the same time) running. Not

7/28/2019 icpr2011

34/436


only tense, but also predicates such as dead ampliate the subject. Someman is dead is true because something which was a man is now dead.

Modal verbs also ampliate their subjects. But there was disagreementhow they did so, and what the truth-conditions of modal propositions were.

For example, Cars can run on hydrogen is true even if no existing cars canrun on hydrogen provided something which could be a car could run on hy-drogen. So the modal verb ampliates the subject to supposit for possiblecars. What of A chimera is conceivable (chimera is ambiguous, but inone sense means an impossible combination of the head of a lion, the bodyof a goat and the tail of a serpent)? Buridan claimed the modal -blehere ampliates only for possibles (so the proposition is false); others, e.g.,Marsilius of Inghen in the next generation, thought such verbs ampliatefor the imaginable, even the impossible (so the proposition is true). Moreproblematic is the supposition of the subject in a proposition of the formEvery S is necessarily P. Buridan claimed that necessarily again ampli-

ates the subject to what is possible, so that Some S might not be P is itscontradictory. William of Ockham disagreed. He eschews the language ofampliation, and thinks that Some S might not be P is ambiguous betweenSomething which is S might not be P and Something which might be Smight not be P, but Every S is necessarily P is not ambiguous, and canonly mean Everything which is S is necessarily P. Thus one reading ofSome S might not be P contradicts Every S is necessarily P, the otherdoes not. Ockham is arguably truer to the everyday understanding of modalpropositions than Buridan, who has a tendency to regiment language to histheory, and in the face of opposition responds that language is a matter ofconvention and he intends to use words the way he wants.

However, none of this explains subalternation. That comes from the the-ory of modes of common personal supposition, that is, of the suppositionof general terms for the things they signify. There are two divisions, intodeterminate and confused supposition, and of confused supposition intoconfused and distributive and merely confused. Broadly, the divisions werecharacterised syntactically in the thirteenth century and semantically in thefourteenth, though accompanied by syntactic rules. Determinate supposi-tion is that of a general term suppositing for many as for one, as do bothterms in Some S is P; confused and distributive that of a term suppositingfor many as for any, as do both terms in No S is P; merely confusedthat of a term like P in Every S is P or in Only Ps are Ss. In confusedand distributive supposition, one can descend (as they termed it) to everysingular, indeed, to a conjunction of singulars, replacing the term in ques-tion by singular terms: Every S is P entails This S is P and that S is P

7/28/2019 icpr2011

35/436

Western Logic 31

and so on for all Ss, so S in the original has confused and distributivesupposition. This kind of descent is invalid for P in Every S is P. Onecan ascend from any singular (so Every S is this P entails Every S isP) but one can only descend through what was called a disjunct term:

Every S is P entails Every S is this P or that P and so on. In determi-nate supposition, one can descend to a disjunction of singulars, and ascendfrom any singular. Thus is subalternation explained: Every S is P entailsThis S is P and that S is P and so on, which in turn entails Some S isP. From confused and distributive supposition to determinate suppositionis valid, but not conversely.

Buridan used the doctrine of supposition, and in particular, the notion ofdistribution in confused and distributive supposition, to provide an alterna-tive to Aristotles explanation of the validity of syllogisms.

It should be noted that by these three conclusions, that is, the

sixth, seventh and eighth, and by the second, the number ofall the modes useful for syllogizing in any of the three figuresboth direct and indirect is made manifest.

The second conclusion showed that nothing follows from two negativepremises, the sixth and seventh that the middle term must be distributed,and the eighth that any term distributed in the conclusion must be dis-tributed in its premise.

4.3 Obligations

Logic lay at the heart of the medieval curriculum, and a further distinc-tive medieval doctrine was the mainstay of the education in logic, that ofobligational disputations. This was a disputation between an Opponentand a Respondent, where the Opponent poses various propositions, as hechooses, and the Respondent is obliged to grant them, deny them or ex-press doubt about them according to closely circumscribed rules hencethe description (logical) obligations. There were several types of obli-gation: let us concentrate on just one, positio. In positio, the Opponentstarts by describing a hypothetical situation and posing (or positing) acertain proposition, the positum. The Respondent must accept it, unlessit is explicitly contradictory; in possible positio, provided it could betrue. E.g., suppose as hypothesis that Socrates is not running, and take as

positum, Every man is running. The Respondent accepts this, and the dis-putation now starts. The Opponent proposes a succession of propositions;each proposition is relevant if it follows from (sequens) or is inconsistent

7/28/2019 icpr2011

36/436


with (repugnans) the positum or any proposition previously granted, or thecontradictory of one previously denied; otherwise it is irrelevant. If it isrelevant, the Respondent must grant it if it is sequens and deny it if it isrepugnans; if irrelevant, he must grant it if it is known by the participants

to be true, deny it if known to be false, and express doubt if its truth orfalsity is unknown a classic example is The king is sitting, which isstandardly doubted if irrelevant. Here is a typical sequence of challengeand response:

Opponent Respondent

Suppose Socrates is not runningPositum: Every man is running Accepted (possible)Socrates is running Denied (irrelevant and false)Socrates is a man Denied (relevant and repugnans)

If the Respondent makes a mistake (that is, grants contradictories, or grantsand denies the same proposition) or after a certain agreed time, the dispu-tation ends and an analysis of the disputation ensues.

Not every obligation is as simple as this. Walter Burley, who wrote atreatise on Obligations in 1302 which is usually credited as representing thestandard doctrine, noted that there were certain tricks an Opponent coulduse to force the Respondent to grant any other falsehood compatible withthe positum. For example:

Opponent Respondent

Positum: Every man is running Accepted (possible)

Socrates is not running or youare a bishop Granted (irrelevant and true,since by hypothesis Socrates isnot running)

Socrates is a man Granted (irrelevant and true)Socrates is running Granted (relevant and sequens)You are a bishop Granted (relevant and sequens)

Once one has understood how the Respondent was forced to concede thefalsehood You are a bishop (assuming it is false), one can see that theRespondent can be forced to concede any falsehood whatever.

Like noughts-and-crosses (aka tic-tac-toe), the rules mean that there isalways a winning strategy for the Respondent keeping a clear head, theresponses can be kept consistent. But mistakes are easy, because of theway relevant and irrelevant proposition must be so dierently dealt with.If a positum really is inconsistent, it should not have been accepted to start

7/28/2019 icpr2011

37/436

Western Logic 33

with. Or the disputation may exploit a paradox. Consider this example:

Opponent Respondent

Positum: A man is an ass or

nothing posited is true

Accepted (the second disjunct

could be true)A man is an ass Denied (irrelevant and false)Nothing posited is true Granted (relevant and sequens)The positum is true Granted (relevant and sequens?)Something posited is true Granted (relevant and sequens)Times up.

Contradictories have been granted. So has the Respondent made a mistake?Burley points out that the positum is an insoluble. Insolubles were, inOckhams famous phrase, so called not because they could not be solvedbut because they were dicult to solve. They constitute various kindsof logical paradox, including the Liar paradox itself: What I am saying isfalse. It seems that this cannot be true, since if it were, it would, as it says,be false; and it cannot be false, for if it were, things would be as it says, soit would be true.

4.4 Insolubles

A variety of solutions to the Liar paradox were explored during the me-dieval period. Nine solutions were listed in Thomas Bradwardines treatiseon Insolubles in the early 1320s; fifteen are listed in Paul of Venices Log-ica Magna (The Great Logic) composed during the 1390s. The majorityfall into three classes: the cassationists (cassantes), who claim that nothinghas been said; the restrictionists (restringentes), who claim that no termcan refer to a proposition of which it is part; and those, like Bradwardine,who diagnose a fallacy secundum quid et simpliciter (of relative and ab-solute), following Aristotles comments in Sophistical Refutations ch. 25.The cassationist solution is known almost entirely by report by logicianswho reject the suggestion, only one surviving text, from the early thirteenthcentury, advocating it. The idea is that any attempt to construct a proposi-tion containing a term referring to this very proposition, fails on groundsof circularity to express any sense. More popular, at least before Bradwar-dines devastating criticisms, was the restrictionist solution, sometimes ina naive version, similar to the cassationist story but inferring not that noth-ing had been said, but that the term trying to refer to the proposition ofwhich it is part, in fact must refer to some other proposition of which it isnot part its scope for referring is thus restricted. A more sophisticated

7/28/2019 icpr2011

38/436


version, put forward by Burley and Ockham, among others, proposed thatthe restriction only applied to insolubles, and prevented a term suppositingfor propositions of which it is part, or their contradictories. For exam-ple, Burleys diagnosis of the error in the Respondents responses in the

obligation above was that the proposition The positum is true should notbe granted, for the positum cannot refer to this positum, for part of the

positum contradicts the proposition of which the positum is part. Thepositum must refer to some other positum, so the proposition is irrelevantand should be responded to according to what holds of that positum. In anycase, contradiction is avoided.

Bradwardine attacked the restrictionist view mercilessly, pointing outhow implausible it was. His own view was taken up directly by very few(Ralph Strode, writing a generation later in the 1360s, was one of his cham-pions), but he seems to have indirectly influenced most of the later propos-als. The central idea to all these subsequent solutions is that an insoluble

says more than appears on the surface. For whatever reason (and the rea-sons were multifarious), an insoluble like What I am saying is false saysnot only that it is false but also that it is true all insolubles, or perhapsall propositions, say implicitly of themselves that they are true. Hence noinsoluble can be true, since it is self-contradictory. All insolubles are false.

5 Renaissance to Enlightenment

5.1 The Renaissance

During the fifteenth century a major change came over European logic.Some people have tied this change closely to the French logician PetrusRamus (Pierre de la Rame, 151572), who for his Masters degree in 1536defended the thesis that Everything said by Aristotle is a pack of lies;logic texts that are seen as influenced by Ramus are often referred to asRamist Logic. But there may be a misunderstanding here. As a mastersstudent Ramus may well have been given his thesis title by his teachers so he was being required to defend an obvious falsehood rather in the spiritof the obligational disputations that we described in 4 above. In fact hislogic was not at all anti-aristotelian, but it does illustrate a general trend torelate logic to humanism.

This trend can be traced back earlier than Ramus. In fact some of its mainfeatures are already visible in the colourful Majorcan eccentric RamonLlull (c. 1300), who proposed to use logic as a tool for converting NorthAfrican Muslims to Christianity. Llull seems to have had little influence in

7/28/2019 icpr2011

39/436

Western Logic 35

his lifetime, but many later logicians have seen his ideas as prophetic. Fourfeatures of his work are worth recording.

First, Llull addressed his logic to the general public, not just to universitystudents and colleagues. (He was deported from Tunis three times for at-

tempting public debates with Muslims there.) During the seventeenth andeighteenth centuries, most publications in logic were for general readers,particularly those with an interest in raising their level of culture. In Britainthe authors were often literary figures rather than university teachers; wehave logic texts from the poets John Milton (17th century), Isaac Watts(18th century) and Samuel Coleridge (early 19th century). Inevitably theseworks avoided all the subtler points of Scholastic logic and said more aboutgeneral improvement of the mind.

Second, Llull wanted to use logic as an instrument of persuasion. In theearly 15th century Lorenzo Valla argued that the central notions of logicshould be not deduction but evidence and testimony; the best logician isone who can present a sound case persuasively. This whole period sawdebates about how to speak both to the heart and to the mind (as BlaisePascal put it in the 17th century). For example one way of catching theinterest of the listener or reader is visual display. Llull himself had somestrange display consisting of rotating disks with Latin words written onthem we will say more on these below. Several writers devised waysof making logic itself more appealing by presenting it as a game; forexample in the 16th century Agostino Nifo wrote a Dialectica Ludicra,which one might translate as Logic by games. This trend towards associ-ating logic with games has become a permanent feature of Western logic.Lewis Carroll joined it in 1887 with his book The Game of Logic. TodayKatalin Havas in Hungary uses games to teach logic to schoolchildren, andJohan van Benthem in the Netherlands does something similar at a moreadvanced level, using some elementary mathematical game theory to ad-vertise epistemic logic. Its worth noting here that in the late 20th centurygame theory was used partly to restore links between logic and probabil-ity, which (as we remarked in 3) were broken when probability theorybecame an independent discipline.

Third, this period saw logic drawing closer to mathematics, in the sensethat logical deductions came to be seen more as calculations. Exactly whatLlull contributed here is unclear, but many later logicians were inspiredby his use of a mechanical device for making logical points. Some peopleeven honour him as a forerunner of computer science. Leibniz named Llullas someone who had anticipated Leibnizs own project (on which morebelow) for building a logical calculus based on mathematics. We should

7/28/2019 icpr2011

40/436


also mention the mathematicians Leonhard Euler (18th century) and JohnVenn (19th century) who gave us respectively Euler Diagrams and VennDiagrams as ways of using visual display to help logical calculations.

The fourth prophetic feature of Llulls approach to logic was his use of

classifications. His rotating disks were meant to illustrate dierent combi-nations of properties from a given set. Leibniz saw this as an anticipationof his own view of logic as a combinatorial art. But it was also an an-ticipation of the enormous interest that some logicians of this period tookin classification and cataloguing. Ramus was famous for his binary classi-fications; in Christopher Marlowes play The Massacre at Paris (c. 1592)Ramus is murdered for being a flat dichotomist. It was during this periodthat notions from Aristotles theory of definition, such as genus, speciesand dierentia, were adapted to provide a structure for biological taxon-omy. Some of the least appealing logic texts of the period are long cata-logues of logical definitions, for example the 236-page Compendium ofChristian Wols Logica published in the mid 18th century by Frobesius.

5.2 Leibniz

The most powerful logician of this period was Gottfried Leibniz (16461716). He was also a mathematician, in fact one of the founders of thedierential and integral calculus. Some of his most lasting contributions tologic are about combining logic and mathematics. He wrote several papersdeveloping a logical calculus of coincidence, i.e. identity. He devised away of translating definitions into numbers, so that logical properties ofthe definitions could be checked by arithmetical calculation. Above all heis remembered for his project of designing a universal characteristic, bywhich he seems to have meant an ideal language in which all human rea-soning can be expressed in a form that can be checked by calculation. Heimagined a day when scholars or lawyers would resolve their dierencesby writing down their arguments in his language and saying to each othercalculemus (let us calculate). The project never came anywhere nearcompletion, but Leibnizs calculi of identity and definitions were certainlyintended to be contributions to it.

It might seem a short step from claiming that all logical proofs can bechecked by calculation, to claiming that all logical problems can be solvedby calculation. Leibniz himself seems never to have taken this step. It wasleft to the 1930s to sort out these claims. By that date, higher-order logichad replaced syllogisms, and a much wider range of logical problems could

7/28/2019 icpr2011

41/436

Western Logic 37

be formulated. Thanks to work of Kurt Gdel and Alan Turing above all(for which see Barry Coopers chapter Computability Theory in this vol-ume), we now know that Leibnizs instincts were sound: for higher-orderlogic we can check by elementary calculation whether a supposed proof

is correct, but by contrast there is no mechanical method of calculationthat will tell us whether any given sentence of the language of higher-orderlogic is a logical truth. The same holds for first-order logic.

Since the mid 20th century, Western modal logicians have often used thenotion of possible worlds: a sentence is necessarily true if and only if it istrue in every possible world. The notion is often credited to Leibniz, whocertainly did talk about alternative worlds that are possible but not actual.But he himself didnt use this notion for logical purposes. In any case onemight argue that the possible worlds of modern modal logicians are notalternative worlds but reference points or viewpoints, as when we say thatsomething will be true at midday tomorrow, or that something is true in

Smiths belief system. (The study of things being true or false at dierenttimes goes back to Aristotle, though Ibn Sna may have been the first tobuild a logic around it. The study of the logic of belief systems is muchmore recent in the West; the Jaina logicians got there first with their notionof perspectives, anekantavada.)

5.3 The philosophical turn

During the late 18th and early 19th centuries several metaphysicians madeattempts to base logic on a theory of rational thinking. The results of theseattempts were strictly not a part of logic at all, but comments on logic

from the outside. But we need to mention them, both because they hadan influence in logic, and because their importance in Western logic hasbeen exaggerated in a number of recent comparisons between Western andIndian logic.

Thus Immanuel Kant (17241804) believed he had identified a centralcore of logic, which he called pure general logic or formal logic. Thedefining feature of pure general logic was that it studies the absolutelynecessary laws of thought without regard to subject matter. All other logicwas dependent on this central core. Some later logicians agreed with Kantthat there is a central genuine logic; in a few cases their definition of it(which was nearly always dierent from Kants) influenced the direction oftheir research, and in this way Kants notion indirectly aected the historyof logic.

Among these later logicians, pride of place goes to Gottlob Frege (1848

7/28/2019 icpr2011

42/436


1925), who aimed to show that arithmetic and mathematical analysis areparts of pure general logic. Frege achieved a combination of depth andprecision that had certainly never been seen in logic before him, and hasrarely been equalled since. But his actual historical influence is another

matter; it is quite subtle to trace and has often been overblown. For exampleone reads that Frege founded mathematical logic; but as we will see insection 6, both the name mathematical logic and its initial programmewere proposed by Giuseppe Peano, quite independently of Frege.

The period around 1800 also saw the formulation of some fundamen-tal laws of thought, such as the Law of Non-Contradiction and the Lawof Excluded Middle. These two laws were popularised in the 1830s bythe Scottish metaphysician William Hamilton in his lectures on logic. For-mulations of the laws have changed over the years, and today few people

would recognise Hamiltons versions. The broad sense of the Law of Non-Contradiction is that it can never be correct both to assert and to deny thesame proposition at the same time. The broad sense of the Law of Ex-cluded Middle is that every proposition either can be correctly asserted orcan be correctly denied (though we might not know which).

The claim that these are fundamental laws bears little relation to tra-ditional practice in Western logic. True, many logicians from Aristotleonwards said things that look like the laws; but one has to allow for sim-plifications and idealisations. In fact many traditional Western logiciansaccepted that a proposition could fail to be straightforwardly true or false

in several circumstances: for example if it was ambiguous, or a border-line case, or paradoxical, or a category mistake. Likewise many traditionallogicians were happy to say that a sentence or proposition (the two were of-ten confused) could be true from one point of view and false from another.Perhaps no Western logician pursued this last point to the same extent asthe Jaina logicians, though Ibn Sna came close at times. In any case, totreat the Laws as a basic dierence between Western and Indian logic iscertainly a distortion.

In the twentieth century it became common to use purpose-built formalsystems of logic. The Laws then served as ways of classifying formalsystems. For example, it is crucial to distinguish Excluded Middle,

Documents

icpr2011